The millionaire and longstanding Vladimir Putin ally said in a video published on Telegram that the handover will be completed by 1 June.
There was no immediate comment from the Russian defence ministry.
It was not possible to verify whether Wagner forces have begun to pull out of the heavily damaged city.
On Wednesday, the Ukrainian General Staff said heavy fighting was continuing inside Bakhmut after a nine-month battle, which has killed tens of thousands of people.
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Ukraine’s deputy defence minister said on Thursday that Wagner units have been replaced by regular troops in the suburbs but Wagner fighters remain in the centre of the city.
Ukrainian forces still have a foothold in the southwestern outskirts, Hanna Maliar said.
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A Wagner victory in Bakhmut would be a much-needed boost for Mr Putin after his invasion lost momentum.
Top Ukrainian presidential adviser Mykhailo Podolyak said Ukraine’s long-awaited counteroffensive is already under way, saying it should not be anticipated as a “single event” starting “at a specific hour of a specific day”.
He said on Twitter that “dozens of different actions to destroy Russian occupation forces” were “taking place yesterday, are taking place today and will continue tomorrow”.
Mr Prigozhin has been in a long-running feud with Moscow and, during the 15-month war, he has repeatedly chastised Russia’s military leadership, accusing them of incompetence and failure to properly supply his troops.
Wagner’s involvement in the capture of Bakhmut has upped Mr Prigozhin’s standing, which he has used to set out his personal views about the war.
“Prigozhin is … using the perception that Wagner is responsible for the capture of Bakhmut to advocate for a preposterous level of influence over the Russian war effort in Ukraine,” according to the Institute for the Study of War.
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3:46
Analysis: Wagner boss appears to insult Putin
His frequent critical commentary about Russia’s military performance is unusual under Mr Putin’s censorious regime.
His latest statement on Bakhmut came a day after he again broke with the Kremlin line on Ukraine.
He said its goal of demilitarising the country has backfired, acknowledged Russian troops have killed civilians and agreed with western estimates that he lost more than 20,000 men in the battle for the city.
Elon Musk is being sued for failing to disclose his purchase of more than 5% of Twitter stock in a timely fashion.
The world’s richest man bought the stock in March 2022 and the complaint by the US Securities and Exchange Commission (SEC) said the delay allowed him to continue buying Twitter stock at artificially low prices.
In papers filed in Washington DC federal court, the SEC said the move allowed Mr Musk to underpay by at least $150m (£123m).
The commission wants Mr Musk to pay a civil fine and give up profits he was not entitled to.
In response to the lawsuit a lawyer for the multi-billionaire said: “Mr Musk has done nothing wrong and everyone sees this sham for what it is.”
An SEC rule requires investors to disclose within 10 calendar days when they cross a 5% ownership threshold.
The SEC said Mr Musk did not disclose his state until 4 April 2022, 11 days after the deadline – by which point he owned more than 9% of Twitter’s shares.
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Twitter’s share price rose by more than 27% following Mr Musk’s disclosure, the SEC added.
Mr Musk later purchased Twitter for $44bn (£36bn) in October 2022 and renamed the social media site X.
Since the election of Donald Trump, Mr Musk has been put in charge of leading a newly created Department of Government Efficiency (DOGE) alongside former Republican presidential candidate Vivek Ramaswamy.
The president-elect said the department would work to reduce government bureaucracy, slash excess regulations, cut wasteful expenditures and restructure federal agencies.
US president-elect Donald Trump has suggested Israel and Hamas could agree a Gaza ceasefire by the end of the week.
Talks between Israeli and Hamas representatives resumed in the Qatari capital Doha yesterday, after US President Joe Biden indicated a deal to stop the fighting was “on the brink” on Monday.
A draft agreement has been sent to both sides. It includes provisions for the release of hostages and a phased Israeli troop withdrawal from Gaza.
Qatar says Israel and Hamas are at their “closest point” yet to a ceasefire deal.
Two Hamas officials said the group has accepted the draft agreement, with Israel still considering the deal.
An Israeli official said a deal is close but “we are not there” yet.
More than 46,500 Palestinians have been killed in Gaza since Israel launched its ground offensive in the aftermath of the 7 October attacks, according to the territory’s Hamas-run health ministry.
President Biden said it would include a hostage release deal and a “surge” of aid to Palestinians, in his final foreign policy speech as president.
“So many innocent people have been killed, so many communities have been destroyed. Palestinian people deserve peace,” he said.
“The deal would free the hostages, halt the fighting, provide security to Israel, and allow us to significantly surge humanitarian assistance to the Palestinians who suffered terribly in this war that Hamas started.”
Qatari mediators have sent Israel and Hamas a draft proposal for an agreement to halt the fighting.
President-elect Donald Trump has also discussed a possible peace deal during a phone interview with the Newsmax channel.
“We’re very close to getting it done and they have to get it done,” he said.
“If they don’t get it done, there’s going to be a lot of trouble out there, a lot of trouble, like they have never seen before.
“And they will get it done. And I understand there’s been a handshake and they’re getting it finished and maybe by the end of the week. But it has to take place, it has to take place.”
Israeli official: Former Hamas leader held up deal
Speaking on Tuesday as negotiations resumed in Qatar, an anonymous Israeli official said that an agreement was “close, but we are not there”.
They accused Hamas of previously “dictating, not negotiating” but said this has changed in the last few weeks.
“Yahya Sinwar was the main obstacle for a deal,” they added.
Sinwar, believed to be the mastermind of the 7 October attacks, led Hamas following the assassination of his predecessor but was himself killed in October last year.
Under Sinwar, the Israeli official claimed, Hamas was “not in a rush” to bring a hostage deal but this has changed since his death and since the IDF “started to dismantle the Shia axis”.
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1:14
Biden: ‘Never, never, never, ever give up’
Iran ‘weaker than it’s been in decades’
Yesterday, President Biden also hailed Washington’s support for Israel during two Iranian attacks in 2024.
“All told, Iran is weaker than it’s been in decades,” the president said.
Mr Biden claimed America’s adversaries were weaker than when he took office four years ago and that the US was “winning the worldwide competition”.
“Compared to four years ago, America is stronger, our alliances are stronger, our adversaries and competitors are weaker,” he said.
“We have not gone to war to make these things happen.”
The US president is expected to give a farewell address on Wednesday.
The deal would see a number of things happen in a first stage, with negotiations for the second stage beginning in the third week of the ceasefire.
It would also allow a surge in humanitarian aid into Gaza, which has been devastated by more than a year of war.
Details of what the draft proposal entails have been emerging on Tuesday, reported by Israeli and Palestinian officials.
Hostages to be returned
In the first stage of the potential ceasefire, 33 hostages would be set free.
These include women (including female soldiers), children, men over the age of 50, wounded and sick.
Israelbelieves most of these hostages are alive but there has not been any official confirmation from Hamas.
In return for the release of the hostages, Israel would free more than 1,000 Palestinian prisoners and detainees.
People serving long sentences for deadly attacks would be included in this but Hamas fighters who took part in the 7 October attack would not be released.
An arrangement to prevent Palestinian “terrorists” from going back to the West Bank would be included in the deal, an anonymous Israeli official said.
The agreement also includes a phased withdrawal of Israeli forces from Gaza, with IDF troops remaining in the border perimeter to defend Israeli border towns and villages.
Security arrangements would be implemented at the Philadelphi corridor – a narrow strip of land that runs along the border between Egypt and Gaza – with Israel withdrawing from parts of it after the first few days of the deal.
The Rafah Crossing between Egypt and Gaza would start to work gradually to allow the crossing of people who are sick and other humanitarian cases out of Gaza for treatment.
Unarmed North Gaza residents would be allowed to return to their homes, with a mechanism introduced to ensure no weapons are moved there.
“We will not leave the Gaza Strip until all our hostages are back home,” the Israeli official said.
What will happen to Gaza in the future?
There is less detail about the future of Gaza – from how it will be governed, to any guarantees that this agreement will bring a permanent end to the war.
“The only thing that can answer for now is that we are ready for a ceasefire,” the Israeli official said.
“This is a long ceasefire and the deal that is being discussed right now is for a long one. There is a big price for releasing the hostages and we are ready to pay this price.”
The international community has said Gaza must be run by Palestinians, but there has not been a consensus about how this should be done – and the draft ceasefire agreement does not seem to address this either.
In the past, Israel has said it will not end the war leaving Hamas in power. It also previously rejected the possibility of the Palestinian Authority, which exercises limited governing powers in the West Bank, from taking over the administration of Gaza.
Since the beginning of its military campaign in Gaza, Israel has also said it would retain security control over the territory after the fighting ends.